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            <ul>
<li><a class="reference internal" href="#">Partial Dependence Plots</a><ul>
<li><a class="reference internal" href="#california-housing-data-preprocessing">California Housing data preprocessing</a></li>
<li><a class="reference internal" href="#partial-dependence-computation-for-multi-layer-perceptron">Partial Dependence computation for multi-layer perceptron</a></li>
<li><a class="reference internal" href="#partial-dependence-computation-for-gradient-boosting">Partial Dependence computation for Gradient Boosting</a></li>
<li><a class="reference internal" href="#analysis-of-the-plots">Analysis of the plots</a></li>
<li><a class="reference internal" href="#d-interaction-plots">3D interaction plots</a></li>
</ul>
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  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-inspection-plot-partial-dependence-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
</div>
<div class="sphx-glr-example-title section" id="partial-dependence-plots">
<span id="sphx-glr-auto-examples-inspection-plot-partial-dependence-py"></span><h1>Partial Dependence Plots<a class="headerlink" href="#partial-dependence-plots" title="Permalink to this headline">¶</a></h1>
<p>Partial dependence plots show the dependence between the target function <a class="footnote-reference brackets" href="#id4" id="id1">2</a>
and a set of ‘target’ features, marginalizing over the values of all other
features (the complement features). Due to the limits of human perception, the
size of the target feature set must be small (usually, one or two) thus the
target features are usually chosen among the most important features.</p>
<p>This example shows how to obtain partial dependence plots from a
<a class="reference internal" href="../../modules/generated/sklearn.neural_network.MLPRegressor.html#sklearn.neural_network.MLPRegressor" title="sklearn.neural_network.MLPRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">MLPRegressor</span></code></a> and a
<a class="reference internal" href="../../modules/generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a> trained on the
California housing dataset. The example is taken from <a class="footnote-reference brackets" href="#id3" id="id2">1</a>.</p>
<p>The plots show four 1-way and two 1-way partial dependence plots (omitted for
<a class="reference internal" href="../../modules/generated/sklearn.neural_network.MLPRegressor.html#sklearn.neural_network.MLPRegressor" title="sklearn.neural_network.MLPRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">MLPRegressor</span></code></a> due to computation time). The
target variables for the one-way PDP are: median income (<code class="docutils literal notranslate"><span class="pre">MedInc</span></code>), average
occupants per household (<code class="docutils literal notranslate"><span class="pre">AvgOccup</span></code>), median house age (<code class="docutils literal notranslate"><span class="pre">HouseAge</span></code>), and
average rooms per household (<code class="docutils literal notranslate"><span class="pre">AveRooms</span></code>).</p>
<dl class="footnote brackets">
<dt class="label" id="id3"><span class="brackets"><a class="fn-backref" href="#id2">1</a></span></dt>
<dd><p>T. Hastie, R. Tibshirani and J. Friedman, “Elements of Statistical
Learning Ed. 2”, Springer, 2009.</p>
</dd>
<dt class="label" id="id4"><span class="brackets"><a class="fn-backref" href="#id1">2</a></span></dt>
<dd><p>For classification you can think of it as the regression score before
the link function.</p>
</dd>
</dl>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="kn">from</span> <span class="nn">time</span> <span class="kn">import</span> <span class="n">time</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">Axes3D</span>

<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">QuantileTransformer</span>
<span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">make_pipeline</span>

<span class="kn">from</span> <span class="nn">sklearn.inspection</span> <span class="kn">import</span> <span class="n">partial_dependence</span>
<span class="kn">from</span> <span class="nn">sklearn.inspection</span> <span class="kn">import</span> <span class="n">plot_partial_dependence</span>
<span class="kn">from</span> <span class="nn">sklearn.experimental</span> <span class="kn">import</span> <span class="n">enable_hist_gradient_boosting</span>  <span class="c1"># noqa</span>
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">HistGradientBoostingRegressor</span>
<span class="kn">from</span> <span class="nn">sklearn.neural_network</span> <span class="kn">import</span> <span class="n">MLPRegressor</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">fetch_california_housing</span>
</pre></div>
</div>
<div class="section" id="california-housing-data-preprocessing">
<h2>California Housing data preprocessing<a class="headerlink" href="#california-housing-data-preprocessing" title="Permalink to this headline">¶</a></h2>
<p>Center target to avoid gradient boosting init bias: gradient boosting
with the ‘recursion’ method does not account for the initial estimator
(here the average target, by default)</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">cal_housing</span> <span class="o">=</span> <span class="n">fetch_california_housing</span><span class="p">()</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">cal_housing</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">cal_housing</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">cal_housing</span><span class="o">.</span><span class="n">target</span>

<span class="n">y</span> <span class="o">-=</span> <span class="n">y</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>

<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
                                                    <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="partial-dependence-computation-for-multi-layer-perceptron">
<h2>Partial Dependence computation for multi-layer perceptron<a class="headerlink" href="#partial-dependence-computation-for-multi-layer-perceptron" title="Permalink to this headline">¶</a></h2>
<p>Let’s fit a MLPRegressor and compute single-variable partial dependence
plots</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Training MLPRegressor...&quot;</span><span class="p">)</span>
<span class="n">tic</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="n">est</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span><span class="n">QuantileTransformer</span><span class="p">(),</span>
                    <span class="n">MLPRegressor</span><span class="p">(</span><span class="n">hidden_layer_sizes</span><span class="o">=</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span> <span class="mi">50</span><span class="p">),</span>
                                 <span class="n">learning_rate_init</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span>
                                 <span class="n">early_stopping</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
<span class="n">est</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;done in </span><span class="si">{:.3f}</span><span class="s2">s&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">tic</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test R2 score: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">est</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">)))</span>
</pre></div>
</div>
<p>We configured a pipeline to scale the numerical input features and tuned the
neural network size and learning rate to get a reasonable compromise between
training time and predictive performance on a test set.</p>
<p>Importantly, this tabular dataset has very different dynamic ranges for its
features. Neural networks tend to be very sensitive to features with varying
scales and forgetting to preprocess the numeric feature would lead to a very
poor model.</p>
<p>It would be possible to get even higher predictive performance with a larger
neural network but the training would also be significantly more expensive.</p>
<p>Note that it is important to check that the model is accurate enough on a
test set before plotting the partial dependence since there would be little
use in explaining the impact of a given feature on the prediction function of
a poor model.</p>
<p>Let’s now compute the partial dependence plots for this neural network using
the model-agnostic (brute-force) method:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Computing partial dependence plots...&#39;</span><span class="p">)</span>
<span class="n">tic</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="c1"># We don&#39;t compute the 2-way PDP (5, 1) here, because it is a lot slower</span>
<span class="c1"># with the brute method.</span>
<span class="n">features</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;MedInc&#39;</span><span class="p">,</span> <span class="s1">&#39;AveOccup&#39;</span><span class="p">,</span> <span class="s1">&#39;HouseAge&#39;</span><span class="p">,</span> <span class="s1">&#39;AveRooms&#39;</span><span class="p">]</span>
<span class="n">plot_partial_dependence</span><span class="p">(</span><span class="n">est</span><span class="p">,</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">features</span><span class="p">,</span>
                        <span class="n">n_jobs</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">grid_resolution</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;done in </span><span class="si">{:.3f}</span><span class="s2">s&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">tic</span><span class="p">))</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">gcf</span><span class="p">()</span>
<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Partial dependence of house value on non-location features</span><span class="se">\n</span><span class="s1">&#39;</span>
             <span class="s1">&#39;for the California housing dataset, with MLPRegressor&#39;</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">hspace</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="partial-dependence-computation-for-gradient-boosting">
<h2>Partial Dependence computation for Gradient Boosting<a class="headerlink" href="#partial-dependence-computation-for-gradient-boosting" title="Permalink to this headline">¶</a></h2>
<p>Let’s now fit a GradientBoostingRegressor and compute the partial dependence
plots either or one or two variables at a time.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Training GradientBoostingRegressor...&quot;</span><span class="p">)</span>
<span class="n">tic</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="n">est</span> <span class="o">=</span> <span class="n">HistGradientBoostingRegressor</span><span class="p">()</span>
<span class="n">est</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;done in </span><span class="si">{:.3f}</span><span class="s2">s&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">tic</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test R2 score: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">est</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">)))</span>
</pre></div>
</div>
<p>Here, we used the default hyperparameters for the gradient boosting model
without any preprocessing as tree-based models are naturally robust to
monotonic transformations of numerical features.</p>
<p>Note that on this tabular dataset, Gradient Boosting Machines are both
significantly faster to train and more accurate than neural networks. It is
also significantly cheaper to tune their hyperparameters (the default tend to
work well while this is not often the case for neural networks).</p>
<p>Finally, as we will see next, computing partial dependence plots tree-based
models is also orders of magnitude faster making it cheap to compute partial
dependence plots for pairs of interacting features:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Computing partial dependence plots...&#39;</span><span class="p">)</span>
<span class="n">tic</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="n">features</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;MedInc&#39;</span><span class="p">,</span> <span class="s1">&#39;AveOccup&#39;</span><span class="p">,</span> <span class="s1">&#39;HouseAge&#39;</span><span class="p">,</span> <span class="s1">&#39;AveRooms&#39;</span><span class="p">,</span>
            <span class="p">(</span><span class="s1">&#39;AveOccup&#39;</span><span class="p">,</span> <span class="s1">&#39;HouseAge&#39;</span><span class="p">)]</span>
<span class="n">plot_partial_dependence</span><span class="p">(</span><span class="n">est</span><span class="p">,</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">features</span><span class="p">,</span>
                        <span class="n">n_jobs</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">grid_resolution</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;done in </span><span class="si">{:.3f}</span><span class="s2">s&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">tic</span><span class="p">))</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">gcf</span><span class="p">()</span>
<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Partial dependence of house value on non-location features</span><span class="se">\n</span><span class="s1">&#39;</span>
             <span class="s1">&#39;for the California housing dataset, with Gradient Boosting&#39;</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">wspace</span><span class="o">=</span><span class="mf">0.4</span><span class="p">,</span> <span class="n">hspace</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="analysis-of-the-plots">
<h2>Analysis of the plots<a class="headerlink" href="#analysis-of-the-plots" title="Permalink to this headline">¶</a></h2>
<p>We can clearly see that the median house price shows a linear relationship
with the median income (top left) and that the house price drops when the
average occupants per household increases (top middle).
The top right plot shows that the house age in a district does not have
a strong influence on the (median) house price; so does the average rooms
per household.
The tick marks on the x-axis represent the deciles of the feature values
in the training data.</p>
<p>We also observe that <a class="reference internal" href="../../modules/generated/sklearn.neural_network.MLPRegressor.html#sklearn.neural_network.MLPRegressor" title="sklearn.neural_network.MLPRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">MLPRegressor</span></code></a> has much
smoother predictions than
<a class="reference internal" href="../../modules/generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a>. For the plots to be
comparable, it is necessary to subtract the average value of the target
<code class="docutils literal notranslate"><span class="pre">y</span></code>: The ‘recursion’ method, used by default for
<a class="reference internal" href="../../modules/generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a>, does not account
for the initial predictor (in our case the average target). Setting the
target average to 0 avoids this bias.</p>
<p>Partial dependence plots with two target features enable us to visualize
interactions among them. The two-way partial dependence plot shows the
dependence of median house price on joint values of house age and average
occupants per household. We can clearly see an interaction between the
two features: for an average occupancy greater than two, the house price is
nearly independent of the house age, whereas for values less than two there
is a strong dependence on age.</p>
</div>
<div class="section" id="d-interaction-plots">
<h2>3D interaction plots<a class="headerlink" href="#d-interaction-plots" title="Permalink to this headline">¶</a></h2>
<p>Let’s make the same partial dependence plot for the 2 features interaction,
this time in 3 dimensions.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>

<span class="n">features</span> <span class="o">=</span> <span class="p">(</span><span class="s1">&#39;AveOccup&#39;</span><span class="p">,</span> <span class="s1">&#39;HouseAge&#39;</span><span class="p">)</span>
<span class="n">pdp</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">partial_dependence</span><span class="p">(</span><span class="n">est</span><span class="p">,</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">features</span><span class="o">=</span><span class="n">features</span><span class="p">,</span>
                               <span class="n">grid_resolution</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
<span class="n">XX</span><span class="p">,</span> <span class="n">YY</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">pdp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">T</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">Axes3D</span><span class="p">(</span><span class="n">fig</span><span class="p">)</span>
<span class="n">surf</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">XX</span><span class="p">,</span> <span class="n">YY</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">rstride</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">cstride</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
                       <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">BuPu</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="n">features</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="n">features</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s1">&#39;Partial dependence&#39;</span><span class="p">)</span>
<span class="c1">#  pretty init view</span>
<span class="n">ax</span><span class="o">.</span><span class="n">view_init</span><span class="p">(</span><span class="n">elev</span><span class="o">=</span><span class="mi">22</span><span class="p">,</span> <span class="n">azim</span><span class="o">=</span><span class="mi">122</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">surf</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">&#39;Partial dependence of house value on median</span><span class="se">\n</span><span class="s1">&#39;</span>
             <span class="s1">&#39;age and average occupancy, with Gradient Boosting&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">top</span><span class="o">=</span><span class="mf">0.9</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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